WKB Computations on Morse Potential

The semiclassical Wentzel–Kramers–Brillouin (WKB) method applied to one-dimensional problems with bound states often reduces to the Sommerfeld–Wilson quantization conditions, the cyclic phase-space integrals . It turns out that this formula gives the exact bound-state energies for the Morse oscillator with . The requisite integral can be reduced to , in which and are the classical turning points . The integral can be done "by hand", using the transformation followed by a contour integration in the complex plane, but Mathematica can evaluate the integral explicitly, needing only the additional fact that The result reads, which can be solved for to give , , in units with . The highest bound state is given by , where represents the floor, which for positive numbers is simply the integer part. The values of , , and (expressed in atomic units) used in this Demonstration are for illustrative purposes only and are not necessarily representative of any actual diatomic molecule.

THINGS TO TRY

SNAPSHOTS

DETAILS

A particle in a one-dimensional potential can be described by the Schrödinger equation: .

The semiclassical or WKB method is based on the ansatz . In the limit as , in the exponential satisfies the Hamilton–Jacobi equation for the action function , with one solution . It is then shown in most graduate-level texts on quantum mechanics (e.g. Schiff, Merzbacher, etc.) that this usually leads to the Sommerfeld–Wilson quantum conditions on periodic orbits , . For one-dimensional problems, the cyclic integral can be replaced by , where , are the classical turning points of the motion and .

As an étude consider the linear harmonic oscillator, with and . The quantum condition reads , which can be solved to give . This is one of a small number of cases in which the WKB method gives the exact quantum-mechanical energies.